On Some Characterization of Preinvex Fuzzy Mappings

In this paper, a new notion of generalized convex fuzzy mapping is introduced, which is called -preinvex fuzzy mapping on the -invex set. We have investigated the characterization of preinvex fuzzy mappings using -preinvex fuzzy mappings, which can be viewed as a novel and innovative application. Some important and significant special cases are discussed. We have also investigated that the minimum of -preinvex fuzzy mappings can be characterized by fuzzy -variational like inequalities.


Introduction
The concept of invex set was firstly introduced by Israel and Mond [2], in 1986. Dealing with convexity they made a comparative study that investigated some different properties of invex sets. In convex set ⊆ ℝ , the path 1 − + = + − which is contained in , reflect that the starting and end point of path is and , respectively. If − = , , then convex set becomes invex set ⊆ ℝ with respect to bi-function : × ⟶ ℝ because + − = + , . But the converse is not true because the path essentially says that path starting from which is included in and do not need that have to be one of the end point of the path which is contained in convexity. Convexity plays an essential role in many areas of mathematical analysis and due to its vast applications in diverse areas, many authors 18 extensively generalized and extended this concept using novel and different approaches. In [9], Hasnon introduced a useful generalization of convex function that is invex function and proved the validity of significant results that hold both for convex and invex function under few conditions. Israel and Mond [2], introduced the concept of preinvex functions on the invex sets. Later on Mohan and Neogy [12], further extended their work by proving that subject to certain conditions a preinvex function defined on the invex set is invex function and vice versa and they also showed that quasi-invex function is quasipreinvex function. Noor [18], and Weir and Mond [28], showed that the preinvex functions preserved some important properties of convex functions. Furthermore, Noor [15], and Yang and Chen [30], studied the optimality conditions of differentiable preinvex functions on the invex set that can be characterized by variational inequalities. Another class of -preinvex mapping on the -invex set (non-convex function) was introduced by Jeyakumar and Mond [10], that have a significant applications in generalized convex programming and multiobjective optimization. In [5], Chang and Zadeh introduced the fuzzy mappings. Many authors, Furukawa [7], Nanda, and Kar [13], and Syau [27], worked on the concept of fuzzy mapping from ℝ to the set of fuzzy numbers, Lipschitz continuity of fuzzy valued, fuzzy logarithmic convex and quasiconvex fuzzy mappings. Based on the concept of ordering illustrated by Goetschel and Voxman [8], Yan and Xu [31], presented the concepts of epigraphs and convexity of fuzzy mappings, and described the characteristic of convex fuzzy and quasi-convex fuzzy mappings. The idea of fuzzy convexity has been generalized and extended in diversity of directions, which has significant implementation in many areas. It is worthy to mention one of the most considered generalization of convex fuzzy mapping is preinvex fuzzy mapping. The idea of fuzzy preinvex mapping on the fuzzy invex set was introduced and studied by Noor [14], and verified that a fuzzy optimality conditions of differentiable fuzzy preinvex mappings can be distinguished by variational-like inequalities. Moreover, any local minimum of a preinvex fuzzy mapping is a global minimum on invex set and necessary and sufficient condition for fuzzy mapping is to be preinvex if its epigraph is an invex set. In [22], Syau further modified the concept of preinvex fuzzy mapping that was presented by Noor [14]. Syau and Lee [26], also discussed the terminologies of continuity and convexity through linear ordering and metric defined on fuzzy numbers. Extension in the Weirstrass Theorem from real-valued functions to fuzzy mappings is also one of their significant contribution in the literature. Li and Noor [11], established an equivalence condition of preinvex fuzzy mapping and characterizations about preinvex fuzzy mappings with some conditions. With the support of examples, Wu and Xu [29],  [19], reviewed the presented literature and made necessary amendments in the results presented by Wu and Xu [29], about invex fuzzy mappings. Rufian-Lizana et al. [20], provided the necessary and sufficient condition for differentiable and twice differentiable preinvex fuzzy mappings. With the help of examples, they proved validity of characterizations and improved previous result given with the help if some conditions by Li and Noor [11]. For further study of literature, we refer to reader about applications and properties of the variational-like inequalities and generalized convex fuzzy mappings, see [1,3,4,6,16,17,21,22,23,24,25], and the references therein Motivated and inspired by the ongoing research work and by the importance of the idea of invexity and preinvexity of mappings. In Section 2, we review some basic definitions, preliminary notations and results. The main results are considered and discussed in Section 3. We have characterized preinvex fuzzy mappings in terms of -preinvex fuzzy mappings. In Section 3.1, the notions of -preinvex, quasi -preinvex and log -preinvex fuzzy mappings are introduced and some properties are investigated. In Section 3.2, we introduce several new concepts of -invex fuzzy mappings and -monotonicities fuzzy operators and then discuss their relation. In Section 4, we have shown that the minimum of -preinvex fuzzy mappings can be distinguished by fuzzy -variational like inequalities which is itself an interesting outcome of our main results.
Definition 5 (see [4]). A fuzzy mapping X Y : → 4 & is said to be convex on the convex set if

Strictly convex fuzzy mapping if strict inequality holds for
Definition 6 (see [4]). A fuzzy mapping X Y : → 4 & is said to be quasi-convex on the convex set if Definition 7 (see [2]). The set in ℝ is said to be invex set with respect to (with respect to) arbitrary bi-function . , . , if The invex set is also called -connected set. Note that, convex set , = − is called an invex set in classical sence, but the converse is not valid.
Definition 8 (see [22]). A fuzzy mapping X Y : → 4 & is said to be preinvex on invex set with respect to bi-function if where : × → ℝ.
Strictly preinvex fuzzy mapping if strict inequality holds for X Y ≠ X Y . X Y : → 4 & is said to be preconcave fuzzy mapping if −X Y is preinvex on . Strictly preconcave fuzzy mapping if strict inequality holds for X Y ≠ X Y .
Definition 9 (see [22]). The fuzzy mapping X Y : → 4 & is said to be quasi-preinvex on invex set with respect to if Definition 10 (see [14]). A fuzzy mapping X Y : → 4 & is said to be log-preinvex on invex set with respect to bi-function if there exist a positive number W such that where X Y . ≻ 0 Y .
Definition 11 (see [10]). The set o is said to be -invex set with respect to arbitrary bifunctions . , . and . , . , if The -invex set o is also called -connected set. Note that, convex set with , = 1 and , = − is called an invex set in classical sense, but the converse is not valid. For example, see [12],

Main Results
Let o be a nonempty -invex subset of ℝ with respect to , . Let X Y : o → 4 & be continuous mapping and : o × o → ℝ be an arbitrary continuous bi-function. Let : o × o → ℝ\0 be a bi-function. We denote ‖. ‖ and 〈. , . 〉 the norm and inner product, respectively.

x-preinvex fuzzy mappings
Definition 12. Let o be a -invex set with respect to , . Then fuzzy mapping X Y : o → 4 & is said to be -preinvex with respect to bi-functions . , . and . , . if Strictly -preinvex fuzzy mapping if strict inequality holds for X Y ≠ X Y and X Y : o → 4 & is said to be -preconcave fuzzy mapping if −X Y is -preinvex on o . Strictly -preconcave fuzzy mapping if strict inequality holds for X Y ≠ X Y . Note that every convex fuzzy mapping with , = 1 and , = − is called preinvex fuzzy mapping but converse does not hold, see example 1.
(v) If = > e , then (3.1) becomes The fuzzy mapping X Y is called the J--preinvex. For , = 1, 3.2 reduces to then fuzzy mapping X Y is called the J-preinvex.
We also define the affine J--preinvex mapping.

Definition 13.
A fuzzy mapping X Y : → 4 & is said to be affine -preinvex on the -invex set o with respect to , , if Note that, if a fuzzy mapping is both -preinvex and -preconcave, then it is a affine -preinvex fuzzy mapping with respect to , .

If = > e
, then fuzzy mapping X Y is called a quadratic equation with respect to , such that This fuzzy mapping is also called affine J--preinvex.
Hence, X Y is strictly quasi -preinvex fuzzy mapping with respect to , . where X Y . ≻ 0 Y .
Similarly, a fuzzy mapping X Y is said to be log -preconcave if −X Y is log -preinvex on o .
If , = 1, then (3.6) becomes The mapping X Y is log-preinvex fuzzy mapping with respect to .
From Definition 16, we have

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It can easily be seen that log -preinvex fuzzy mapping ⇒ -preinvex fuzzy mapping ⇒ quasi -preinvex fuzzy mapping. For = 1, Definition 12 and Definition 16, reduces to: Which plays an important role in studying the properties of -preinvex fuzzy mappings and -invex fuzzy mappings. If , = 1, then Condition A reduces to the following for preinvex fuzzy mappings.
For the applications of Condition B, see [14,16,20,23]. Proof. Let X Y ≺ X Y and X Y be a pseudo -preinvex fuzzy mapping. Then where • , = X Y − X Y . This prove that X Y is pseudo -preinvex fuzzy mapping.

G-differentiable -preinvex fuzzy mappings
In this section, we have proposed the concepts of -invex fuzzy mappings andmonotone fuzzy operators. With the support these notions, we have investigated some properties of G-differentiable -preinvex fuzzy mappings. Hence, the result follows.
We need the following assumption regarding the bi-function , which plays an important role in G-differentiation of the main results.
It is well known that, each G-differentiable preinvex fuzzy mapping is invex fuzzy mapping but to prove its converse we need special condition.
It can easily be seen that, if , = 1, then Condition C collapse to the following condition which is due to Neogy and Mohen [13].
The required (4.1), since X Y is a G-differentiable fuzzy mapping.
Conversely, let ∈ o satisfies (4.1). Since X Y is a G-differentiable -preinvex fuzzy mapping, then by (3.7), we have X Y − X Y ⪰ 〈X Y , , , , 〉, from which, using (21), we have Showing that ∈ o is the minimum of the mapping, the required result.
For G-differentiable -invex fuzzy mapping, we have the following result.

Conclusion
In this paper, we have introduced and studied new class of non-convex fuzzy mappings with respect to bi-functions , . Which is called -preinvex fuzzy mappings with respect to , . We have investigated characterization of preinvex fuzzy mappings in term of -preinvex fuzzy mappings. It is shown that convex and preinvex fuzzy mappings are special cases of -preinvex fuzzy mappings. We have proposed several new concepts of -invex fuzzy mappings and -monotonicities, and then discuss their 40 relation. It is proved that minimum of -preinvex mappings can be characterized by -varitional like inequalities and varitional like inequalities. One can obtain important and significant applications in generalized convex fuzzy programming and multiobjective fuzzy optimization, see [10,16].